There exist, OP, FUNCTIONS that may have a divergent series as its argument*, and that function assigns a unique number to that divergent sum that in some forms of technical cases is called a "sum". Borel summation and Abel summation are examples of this.

But, and this is important:
Although such functions can be constructed (and be very useful), and has a number of properties that motivates the use of the "summation" term to designate them, they should not by any standards be CONFUSED with a regular sum.

They are not, they are functions that can have some subclass of divergent series as part of their argument domain.
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*Or, more precisely, having as its argument a sequence of numbers which, if they had been summed in a standard manner, would represent a divergent series

Just because an operation is commutative with a finite number of terms, it does not follow that it is commutative with an infinite number of terms.

Edit: Expanding on this. By the Riemann rearrangement theorem any conditionally convergent series can be rearranged to form any value. Hence commutativity is not necessarily true for convergent series, let alone divergent ones.